Simple Nature - Light and Matter

o/An RC circuit.

exert mechanical forces on things. Modern compact fluorescent

bulbs have ballasts built into their bases that use a frequency in

the kilohertz range, eliminating the flicker and hum.

Discussion Question

A What happens when the physicist at the bottom in figure m/3 starts

getting tired, and decreases the current?

10.5.4 Decay

Up until now I’ve soft-pedaled the fact that by changing the char-

acteristics of an oscillator, it is possible to produce non-oscillatory

behavior. For example, imagine taking the mass-on-a-spring system

and making the spring weaker and weaker. In the limit of small

k, it’s as though there was no spring whatsoever, and the behavior

of the system is that if you kick the mass, it simply starts slowing

down. For friction proportional tov, as we’ve been assuming, the re-

sult is that the velocity approaches zero, but never actually reaches

zero. This is unrealistic for the mechanical oscillator, which will not

have vanishing friction at low velocities, but it is quite realistic in

the case of an electrical circuit, for which the voltage drop across the

resistor really does approach zero as the current approaches zero.

We do not even have to reducekto exactly zero in order to get

non-oscillatory behavior. There is actually a finite, critical value be-

low which the behavior changes, so that the mass never even makes

it through one cycle. This is the case of overdamping, discussed on

page 190.

Electrical circuits can exhibit all the same behavior. For sim-

plicity we will analyze only the cases where either the capacitor or

the inductor is completely absent, givingQ= 0.

The RC circuit

We first analyze the RC circuit, o. In reality one would have

to “kick” the circuit, for example by briefly inserting a battery, in

order to get any interesting behavior. We start with Ohm’s law and

the equation for the voltage across a capacitor:

VR=IR

VC=q/C

The loop rule tells us

VR+VC= 0,

and combining the three equations results in a relationship between

qandI:

I=−

1

RC

q

The negative sign tells us that the current tends to reduce the charge

on the capacitor, i.e., to discharge it. It makes sense that the current

622 Chapter 10 Fields